Addition and subtraction of complex numbers

Description of how to add and subtract complex numbers

In the article on complex numbers it was written that the calculation rules for real numbers should also apply to complex numbers. This article describes how to add complex numbers.

As first consider the addition of complex numbers to an example. Suppose we want to add $$\,3 + i\,$$ and $$\,1 - 2i\,$$.

We are looking for $$(3+i)+(1-2i)$$

According to the principle of permanence, the calculation rules of real numbers should continue to be valid. This means that we are doing what we would do with real numbers. We would combine the two real expressions $$3$$ and $$1$$, as well as the two imaginary expressions $$i$$ and $$-2i$$.

That's exactly how we calculate here

$$(3+i)+(1-2i)=3+i+1-2i=(3+1)+(i-2i)=4-i$$

The result of the calculation is $$4 - i$$.

Subtraction

When subtracting a complex number, the real expressions and the imaginary expressions are combined in the same way. It only needs to be noted that the minus sign changes the sign of the second number. You know that about calculating with real numbers.

As an example we subtract the numbers from the example above

$$(3+i)-(1-2i)=3+i-1+2i=(3-1)+(i+2i)=2+3i$$

The result of the calculation is $$2 + 3i$$.

To summarise, we can say, complex numbers are added by adding the real parts and the imaginary parts separately. The same applies to the subtraction. Complex numbers are subtracted by subtracting the real parts and the imaginary parts separately.